Computational Statistics

, Volume 33, Issue 2, pp 709–730 | Cite as

Estimating nonlinear effects in the presence of cure fraction using a semi-parametric regression model

  • Thiago G. Ramires
  • Niel Hens
  • Gauss M. Cordeiro
  • Edwin M. M. Ortega
Original Paper


Nonlinear effects between explanatory and response variables are increasingly present in new surveys. In this paper, we propose a flexible four-parameter semi-parametric cure rate survival model called the sinh Cauchy cure rate distribution. The proposed model is based on the generalized additive models for location, scale and shape, for which any or all parameters of the distribution are parametric linear and/or nonparametric smooth functions of explanatory variables. The new model is used to fit the nonlinear behavior between explanatory variables and cure rate. The biases of the cure rate parameter estimates caused by not incorporating such non-linear effects in the model are investigated using Monte Carlo simulations. We discuss diagnostic measures and methods to select additive terms and their computational implementation. The flexibility of the proposed model is illustrated by predicting lifetime and cure rate proportion as well as identifying factors associated to women diagnosed with breast cancer.


Cure rate models GAMLSS Long-term survivors P-spline Residual analysis 



The first author acknowledge the financial support of the “Ciência sem Fronteiras” program of CNPq (Brazil) under the process number 200574/2015-9.

Compliance with ethical standards

Conflict of interest

The authors have declared no conflict of interest.


  1. Altman DG, Lausen B, Sauerbrei W, Schumacher M (1994) Dangers of using “optimal” cutpoints in the evaluation of prognostic factors. J Natl Cancer Inst 86:829–835CrossRefGoogle Scholar
  2. Balakrishnan N, Pal S (2012) EM algorithm-based likelihood estimation for some cure rate models. J Stat Theory Pract 6:698–724MathSciNetCrossRefGoogle Scholar
  3. Berkson J, Gage RP (1952) Survival curve for cancer patients following treatment. J Am Stat Assoc 47:501–515CrossRefGoogle Scholar
  4. Boag JW (1949) Maximum likelihood estimates of the proportion of patients cured by cancer therapy. J R Stat Soc B 11:15–53zbMATHGoogle Scholar
  5. Buuren SV, Fredriks M (2001) Worm plot: a simple diagnostic device for modelling growth reference curves. Stat Med 20:1259–1277CrossRefGoogle Scholar
  6. Cancho VG, Dey DK, Louzada F (2015) Unified multivariate survival model with a surviving fraction: an application to a Brazilian customer churn data. J Appl Stat 43:572–584MathSciNetCrossRefGoogle Scholar
  7. Chen MH, Ibrahim JG, Sinha D (1999) A new Bayesian model for survival data with a surviving fraction. J Am Stat Assoc 94:909–919MathSciNetCrossRefzbMATHGoogle Scholar
  8. Cooner F, Banerjee S, Carlin BP, Sinha D (2007) Flexible cure rate modeling under latent activation schemes. J Am Stat Assoc 102:560–572MathSciNetCrossRefzbMATHGoogle Scholar
  9. Cordeiro GM, Cancho VG, Ortega EMM, Barriga GDC (2016) A model with long-term survivors: negative binomial Birnbaum–Saunders. Commun Stat Theory Methods 45:1370–1387MathSciNetCrossRefzbMATHGoogle Scholar
  10. da Cruz JN, Ortega EMM, Cordeiro GM (2016) The log-odd log-logistic Weibull regression model: modelling, estimation, influence diagnostics and residual analysis. J Stat Comput Simul 86:1516–1538MathSciNetCrossRefGoogle Scholar
  11. Dunn PK, Smyth GK (1996) Randomized quantile residuals. J Comput Graph Stat 5:236–244Google Scholar
  12. Eilers PH, Marx BD (1996) Flexible smoothing with B-splines and penalties. Stat Sci 11:89–121MathSciNetCrossRefzbMATHGoogle Scholar
  13. Farewell VT (1982) The use of mixture models for the analysis of survival data with long-term survivors. Biometrics 38:1041–1046CrossRefGoogle Scholar
  14. Fitzgibbons PL, Page DL, Weaver D, Thor AD, Allred DC, Clark GM et al (2000) Prognostic factors in breast cancer: College of American Pathologists consensus statement 1999. Arch Pathol Lab Med 124:966–978Google Scholar
  15. Gospodarowicz MK, O’Sullivan B, Sobin LH (eds) (2006) Prognostic factors in cancer. Wiley-Liss, Frankfurt, pp 165–168Google Scholar
  16. Green PJ, Silverman BW (1993) Nonparametric regression and generalized linear models: a roughness penalty approach. CRC Press, Boca RatonzbMATHGoogle Scholar
  17. Hashimoto EM, Ortgea EMM, Cancho VG, Cordeiro GM (2015) A new long-term survival model with interval-censored data. Sankhya B 77:207–239MathSciNetCrossRefzbMATHGoogle Scholar
  18. Hashimoto EM, Cordeiro GM, Ortega EMM, Hamedani GG (2016) New flexible regression models generated by gamma random Variables with censored data. Int J Stat Probab 5:9–31CrossRefGoogle Scholar
  19. Hastie TJ, Tibshirani RJ (1990) Generalized additive models, vol 43. CRC Press, Boca RatonzbMATHGoogle Scholar
  20. Ibrahim JG, Chen MH, Sinha D (2001) Bayesian survival analysis. Springer, New YorkCrossRefzbMATHGoogle Scholar
  21. Ko A (2009) Everyone’s guide to cancer therapy: how cancer is diagnosed, treated, and managed day to day. Andrews McMeel Publishing, Kansas CityGoogle Scholar
  22. Lagakos SW (1988) Effects of mismodelling and mismeasuring explanatory variables on tests of their association with a response variable. Stat Med 7:257–274CrossRefGoogle Scholar
  23. Lanjoni BR, Ortega EMM, Cordeiro GM (2016) Extended Burr XII regression models: theory and applications. J Agric Biol Environ Stat 21:203–224MathSciNetCrossRefzbMATHGoogle Scholar
  24. Lee Y, Nelder JA, Pawitan Y (2006) Generalized linear models with random effects: unified analysis via H-likelihood. CRC Press, Boca RatonCrossRefzbMATHGoogle Scholar
  25. Lønning PE (2007) reast cancer prognostication and prediction: are we making progress? Ann Oncol 18(suppl 8):viii3–viii7Google Scholar
  26. Maller RA, Zhou X (1996) Survival analysis with long-term survivors. Wiley, New YorkzbMATHGoogle Scholar
  27. Morgan TM, Elashoff RM (1986) Effect of categorizing a continuous covariate on the comparison of survival time. J Am Stat Assoc 81:917–921CrossRefGoogle Scholar
  28. Ortega EMM, Cordeiro GM, Hashimoto EM, Cooray K (2014) A log-linear regression model for the odd Weibull distribution with censored data. J Appl Stat 41:1859–1880MathSciNetCrossRefzbMATHGoogle Scholar
  29. Ortega EMM, Cordeiro GM, Campelo AK, Kattan MW, Cancho VG (2015) A power series beta Weibull regression model for predicting breast carcinoma. Stat Med 34:1366–1388MathSciNetCrossRefGoogle Scholar
  30. Ramires TG, Ortega EMM, Cordeiro GM, Hens N (2016) A bimodal flexible distribution for lifetime data. J Stat Comput Simul 86:2450–2470MathSciNetCrossRefGoogle Scholar
  31. Rigby RA, Stasinopoulos DM (2005) Generalized additive models for location, scale and shape. J R Stat Soc Ser C (Appl Stat) 54:507–554MathSciNetCrossRefzbMATHGoogle Scholar
  32. Rigby RA, Stasinopoulos DM (2014) Automatic smoothing parameter selection in GAMLSS with an application to centile estimation. Stat Methods Med Res 23:318–332MathSciNetCrossRefGoogle Scholar
  33. Stasinopoulos DM, Rigby RA (2007) Generalized additive models for location scale and shape (GAMLSS) in R. J Stat Softw 23:1–46CrossRefGoogle Scholar
  34. Schumacher M, Bastert G, Bojar H, Huebner K et al (1994) Randomized \(2 \times 2\) trial evaluating hormonal treatment and the duration of chemotherapy in node-positive breast cancer patients. German Breast Cancer Study Group. J Clin Oncol 12:2086–2093Google Scholar
  35. Tsodikov AD, Ibrahim JG, Yakovlev AY (2003) Estimating cure rates from survival data: an alternative to two-component mixture models. J Am Stat Assoc 98:1063–1078MathSciNetCrossRefGoogle Scholar
  36. Voudouris V, Gilchrist R, Rigby R, Sedgwick J, Stasinopoulos D (2012) Modelling skewness and kurtosis with the BCPE density in GAMLSS. J Appl Stat 39:1279–1293MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  • Thiago G. Ramires
    • 1
    • 2
  • Niel Hens
    • 2
    • 3
  • Gauss M. Cordeiro
    • 4
  • Edwin M. M. Ortega
    • 5
  1. 1.Department of MathematicsFederal University of Technology - ParanáApucaranaBrazil
  2. 2.Interuniversity Institute for Biostatistics and Statistical Bioinformatics (I-Biostat)University of HasseltHasseltBelgium
  3. 3.Centre for Health Economic Research and Modelling Infectious Diseases, Vaccine and Infectious Disease InstituteUniversity of AntwerpAntwerpenBelgium
  4. 4.Department of StatisticsFederal University of PernambucoRecifeBrazil
  5. 5.Department of Exact SciencesUniversity of São PauloSão PauloBrazil

Personalised recommendations